Application of Inversion of a Matrix in Cryptography - International ...
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
A NOTE ON THE APPLICATION OF QUADRATIC FORMS IN<br />
CODING THEORY WITH A NOTE ON SECURITY.<br />
B<br />
1<br />
2<br />
3<br />
. Vellaikannan<br />
Dr.<br />
V.<br />
Mohan<br />
V.<br />
Gnanaraj<br />
1. Senior Grade Lecturer <strong>in</strong> Mathematics, E Mail : bvkmat@tce.edu, 9944563124, 0452-2562693<br />
Thiagarajar College <strong>of</strong> Eng<strong>in</strong>eer<strong>in</strong>g, Madurai - 625 015. Tamil Nadu , India.<br />
2. Pr<strong>of</strong>essor and HOD <strong>of</strong> Mathematics, E Mail : vmohan@tce.edu, 9894026923, 0452-2482240<br />
Thiagarajar College <strong>of</strong> Eng<strong>in</strong>eer<strong>in</strong>g, Madurai - 625 015, Tamil Nadu , India.<br />
3. Selection Grade Lecturer <strong>in</strong> Mathematics, E Mail : vgmat@tce.edu 9442784592, 0452-2482240<br />
Thiagarajar College <strong>of</strong> Eng<strong>in</strong>eer<strong>in</strong>g, Madurai - 625 015. Tamil Nadu, India.<br />
ABSTRACT<br />
A novel approach which <strong>in</strong>corporates the salient features <strong>of</strong> message shar<strong>in</strong>g<br />
(Called cod<strong>in</strong>g theory) is presented and also extended to the messages <strong>of</strong> higher<br />
length. The proposed method is very simple <strong>in</strong> its pr<strong>in</strong>ciple and has great potential<br />
to be applied to other situations where the exchange <strong>of</strong> messages is done<br />
confidentially.<br />
Keywords<br />
Quadratic forms – Canonical form -- Matrices – <strong>Matrix</strong> Multiplication –<br />
Inverse <strong>Matrix</strong> – Invertible Matrices – Diagonal matrices – <strong>Matrix</strong> <strong>in</strong>duced by<br />
a Quadratic form -Encoder –Decoder – Message <strong>Matrix</strong>.<br />
1. Introduction<br />
Cod<strong>in</strong>g theory is a subject by which the exchange <strong>of</strong> messages is<br />
adm<strong>in</strong>istered <strong>in</strong> a confidential and more secured way hav<strong>in</strong>g a wide application <strong>in</strong> Military<br />
operations, Bank<strong>in</strong>g Transactions etc… Recently there has been a wide application <strong>of</strong><br />
<strong>in</strong>version <strong>of</strong> matrices (<strong>of</strong> order 2 whose <strong>in</strong>verses are readily obta<strong>in</strong>ed) [1], [2] to the<br />
problem <strong>of</strong> exchange <strong>of</strong> messages <strong>in</strong> a confidential and a secured way. In the proposed<br />
method the idea has been extended to the matrices ma<strong>in</strong>ly to nons<strong>in</strong>gular diagonal<br />
matrices <strong>of</strong> higher order, especially <strong>in</strong>duced from Quadratic forms. We know that<br />
determ<strong>in</strong>ation <strong>of</strong> the <strong>in</strong>verse <strong>of</strong> nons<strong>in</strong>gular matrices <strong>of</strong> higher order is difficult and this<br />
requires higher level algorithms for the use <strong>of</strong> computers.<br />
A newly developed method that avoids the difficulties <strong>in</strong> the<br />
determ<strong>in</strong>ation <strong>of</strong> <strong>in</strong>verse <strong>of</strong> a nons<strong>in</strong>gular matrix is <strong>in</strong>troduced <strong>in</strong> this paper. The results<br />
obta<strong>in</strong>ed us<strong>in</strong>g this method works vey well for the whole range <strong>of</strong> message exchang<strong>in</strong>g<br />
problems and the excellent agreement with the exist<strong>in</strong>g one.<br />
1.1 Theorem<br />
A text message <strong>of</strong> str<strong>in</strong>gs <strong>of</strong> some length / size l can be converted <strong>in</strong> to a matrix<br />
(called a message matrix M) <strong>of</strong> size m × n where<br />
n < m and n is the least such that mx n ≥ l depend<strong>in</strong>g up on the length <strong>of</strong> the<br />
message with help <strong>of</strong> suitably chosen numerals and zeros.<br />
78
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
Pro<strong>of</strong> :<br />
The pro<strong>of</strong> is by enumeration on numbers.<br />
Consider the follow<strong>in</strong>g<br />
For the text message <strong>of</strong> length up to l = 9 ; then we have m = 3 ; n = 3.<br />
Similarly For the text message <strong>of</strong> length up to l = 12 we have m = 4 ; n = 3.<br />
And so on. Hence the pro<strong>of</strong>.<br />
2. Basics<br />
Consider the text message<br />
I LOVE VINODHINI<br />
To every letter we will associate a number. The easiest way to do that is to associate 0 to<br />
a blank or space, 1 to A, 2 to B, etc... Another way is to associate 0 to a blank or space, 1<br />
to A, -1 to B, 2 to C, -2 to D, etc... Let us use the second choice. So our message is given<br />
by the str<strong>in</strong>g as<br />
I L O V E V I N O D H I N I<br />
5 0 -6 8 -11 3 0 -11 5 -7 8 -2 -4 5 -7 5<br />
Now we rearrange these numbers <strong>in</strong>to a matrix M (Row wise e/ Coolum wise).we use<br />
column wise. For our case, we have<br />
⎛ 5<br />
⎜<br />
⎜ 0<br />
⎜ − 6<br />
M = ⎜<br />
⎜ 8<br />
⎜<br />
⎜<br />
−11<br />
⎝ 3<br />
0<br />
−11<br />
5<br />
− 7<br />
8<br />
− 2<br />
− 4⎞<br />
⎟<br />
5 ⎟<br />
− 7⎟<br />
⎟<br />
5 ⎟<br />
0<br />
⎟<br />
⎟<br />
0<br />
⎠<br />
<strong>of</strong> order ( 6 × 3)<br />
(Us<strong>in</strong>g Theorem)<br />
Then we perform the product M A , where A is an arbitrary nons<strong>in</strong>gular matrix given by<br />
⎛ −1<br />
5<br />
⎜<br />
A = ⎜−<br />
2 11<br />
⎜<br />
⎝ 1 − 5<br />
Then, we get<br />
−1⎞<br />
⎟<br />
7 ⎟<br />
2 ⎟<br />
⎠<br />
−1<br />
whose <strong>in</strong>verse is given by A =<br />
⎛57<br />
⎜<br />
⎜ 11<br />
⎜<br />
⎝−1<br />
− 5<br />
−1<br />
0<br />
46⎞<br />
⎟<br />
9 ⎟<br />
−1⎟<br />
⎠<br />
79
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
X = MA=<br />
⎛ 5<br />
⎜<br />
⎜ 0<br />
⎜ − 6<br />
⎜<br />
⎜ 8<br />
⎜<br />
⎜<br />
−11<br />
⎝ 3<br />
0<br />
−11<br />
5<br />
− 7<br />
8<br />
− 2<br />
− 4⎞<br />
⎟<br />
5 ⎟<br />
⎟ ⎛ −1<br />
− 7 ⎜<br />
⎟ ⎜−<br />
2<br />
5 ⎟ ⎜<br />
0<br />
⎟ ⎝ 1<br />
⎟<br />
0<br />
⎠<br />
5<br />
11<br />
− 5<br />
−1⎞<br />
⎟<br />
7 ⎟<br />
2 ⎟<br />
⎠<br />
=<br />
⎛ − 9<br />
⎜<br />
⎜ 27<br />
⎜−11<br />
⎜<br />
⎜ 11<br />
⎜<br />
⎜<br />
− 5<br />
⎝ 1<br />
45<br />
−146<br />
60<br />
− 62<br />
33<br />
− 7<br />
−13<br />
⎞<br />
⎟<br />
− 67⎟<br />
27 ⎟<br />
⎟<br />
− 47⎟<br />
67<br />
⎟<br />
⎟<br />
−17<br />
⎠<br />
The encoded numeric message to be sent is<br />
-9, -27, -11, 11,………-47,67,-17.<br />
This encoded message is aga<strong>in</strong> decoded us<strong>in</strong>g the <strong>in</strong>verse <strong>of</strong> A as<br />
M =<br />
−1<br />
XA =<br />
⎛ − 9<br />
⎜<br />
⎜ 27<br />
⎜−11<br />
⎜<br />
⎜ 11<br />
⎜<br />
⎜<br />
− 5<br />
⎝ 1<br />
45<br />
−146<br />
60<br />
− 62<br />
33<br />
− 7<br />
−13<br />
⎞<br />
⎟<br />
− 67⎟<br />
⎟ ⎛57<br />
27 ⎜<br />
⎟ ⎜ 11<br />
− 47⎟<br />
⎜<br />
67<br />
⎟ ⎝−1<br />
⎟<br />
−17<br />
⎠<br />
− 5<br />
−1<br />
0<br />
46⎞<br />
⎟<br />
9 ⎟<br />
−1⎟<br />
⎠<br />
⎛ 5<br />
⎜<br />
⎜ 0<br />
⎜ − 6<br />
= ⎜<br />
⎜ 8<br />
⎜<br />
⎜<br />
−11<br />
⎝ 3<br />
0<br />
−11<br />
5<br />
− 7<br />
8<br />
− 2<br />
− 4⎞<br />
⎟<br />
5 ⎟<br />
− 7⎟<br />
⎟<br />
5 ⎟<br />
0<br />
⎟<br />
⎟<br />
0<br />
⎠<br />
80
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
This matrix is aga<strong>in</strong> converted <strong>in</strong> to a str<strong>in</strong>g <strong>of</strong> numerals as<br />
5 0 -6 8 -11 3 0 -11 5 -7 8 -2 -4 5 -7 5<br />
I L O V E V I N O D H I N I<br />
3. Solution Procedure<br />
A text message <strong>of</strong> str<strong>in</strong>gs <strong>of</strong> some length / size l from the sender is converted <strong>in</strong><br />
to a stream <strong>of</strong> numerals with the help <strong>of</strong> some cod<strong>in</strong>g process (Probably may be the<br />
standard codes like A – 1, B-2,….Z-26 and for space -0 ) which is aga<strong>in</strong> converted <strong>in</strong> to<br />
a matrix (called a message matrix M) <strong>of</strong> size m × n where<br />
n < m and n is the least such that mx n ≥ l depend<strong>in</strong>g up on the length <strong>of</strong> the<br />
message .In such case the size <strong>of</strong> the Encoder (The <strong>in</strong>duced Diagonal matrix <strong>of</strong> a<br />
Quadratic form <strong>of</strong> suitable variables ) becomes n . Then the Encoder need not be an<br />
arbitrary matrix where as it may be taken as a Diagonal matrix <strong>of</strong> size n whose <strong>in</strong>verse<br />
can be readily obta<strong>in</strong>ed.<br />
Then the message matrix is converted <strong>in</strong> to a New <strong>Matrix</strong> X (Encoded <strong>Matrix</strong>)<br />
us<strong>in</strong>g <strong>Matrix</strong> Multiplication as X = ME . Then this is sent to the Receiver. Then the<br />
receiver decode this matrix with the help <strong>of</strong> a matrix D (Decoder matrix) which is<br />
−1<br />
noth<strong>in</strong>g but the <strong>in</strong>verse <strong>of</strong> the encoder ( i . e.,<br />
D = E ) , to get the message matrix back as<br />
−1<br />
M = XE .Then with the previously used codes the receiver can get back the message<br />
<strong>in</strong> terms <strong>of</strong> the numerals which aga<strong>in</strong> can be converted to the orig<strong>in</strong>al text message. When<br />
the length / size <strong>of</strong> the text message is too large, the value <strong>of</strong> n become higher, lead<strong>in</strong>g<br />
to the need <strong>of</strong> higher order diagonal matrices <strong>in</strong>duced from the quadratic forms <strong>of</strong> higher<br />
number <strong>of</strong> variables.<br />
3. 1 Algorithm<br />
3.1.1 Encod<strong>in</strong>g Process<br />
1. Convert the text message <strong>of</strong> length l <strong>in</strong> to a stream <strong>of</strong> Numerals us<strong>in</strong>g a user<br />
friendly scheme for both the .sender and the receiver.<br />
2. Place the numerals <strong>in</strong> to a matrix <strong>of</strong> order m × n where<br />
n < m and n is the least such that mx n ≥ l where n depends on the size<br />
<strong>of</strong> the message and call this as a Message matrix M.<br />
3. Multiply this message matrix by the Encoder E <strong>of</strong> size n. (Normally a <strong>in</strong>duced<br />
diagonal matrix compatible for the product X = ME .) and get the encoded<br />
matrix X.<br />
4. Convert the message matrix <strong>in</strong> to the stream <strong>of</strong> numbers that conta<strong>in</strong>s the<br />
encrypted message and sent to the receiver.<br />
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
3.1 .2 Decod<strong>in</strong>g Process<br />
1. Place the encrypted stream <strong>of</strong> numbers that represent the encrypted message<br />
<strong>in</strong> to a matrix<br />
−1<br />
2. Multiply the encoded matrix X with the decoder D = E (The <strong>in</strong>verse <strong>of</strong> E) to<br />
get back the message matrix M<br />
3. Convert this message matrix <strong>in</strong> to a stream <strong>of</strong> numbers with the help <strong>of</strong> the<br />
orig<strong>in</strong>ally used scheme.<br />
4. Convert this stream <strong>of</strong> numerals <strong>in</strong> to the text <strong>of</strong> the orig<strong>in</strong>al message.<br />
4. Results and Discussions<br />
Consider the message to be sent: BEST WISHES<br />
We take the standard codes as follows:<br />
A →1 ; B → 2 ; ........; Z → 26 and Space → 0<br />
1. We convert the above message <strong>in</strong> to a stream <strong>of</strong> numerical values as<br />
follows:<br />
BEST WISHES 2 5 19 20 0 23 9 19 8 5 19 (Here we admit a s<strong>in</strong>gle<br />
spac<strong>in</strong>g for the purpose better understand<strong>in</strong>g)<br />
2. We construct the message matrix M with this stream <strong>of</strong> numerals as<br />
⎛ 2 5 19⎞<br />
⎜ ⎟<br />
⎜20<br />
0 23⎟<br />
M = ⎜ ⎟ which is <strong>of</strong> order 4x 3.<br />
(Us<strong>in</strong>g Theorem)<br />
9 19 8<br />
⎜ ⎟<br />
⎝ 5 19 0 ⎠<br />
3. Based on this, We take the 3 rd order Diagonal matrix (The<br />
diagonalized matrix <strong>of</strong> the matrix <strong>of</strong> a QF <strong>of</strong> suitable variables<br />
otherwise called the matrix <strong>of</strong> the canonical form) with Diag ( )).<br />
2 2 2<br />
For e:g if the QF is 2x1 + x2<br />
+ x3<br />
+ 2x1x2<br />
− 2x1x3<br />
− 4x2<br />
x3<br />
then the<br />
matrix <strong>of</strong> the QF is<br />
⎛ 2 1 −1⎞<br />
⎜<br />
⎟<br />
2 2 2<br />
⎜ 1 1 − 2⎟<br />
.Also the canonical form is − y<br />
1<br />
+ y2<br />
+ 4y3<br />
whose<br />
⎜<br />
⎟<br />
⎝−1<br />
− 2 1 ⎠<br />
⎛−1<br />
0 0⎞<br />
⎜ ⎟<br />
matrix is given by D ( −1,1,<br />
4 ) = ⎜ 0 1 0⎟<br />
⎜ ⎟<br />
⎝ 0 0 4⎠<br />
⎛−1<br />
0 0⎞<br />
⎜ ⎟<br />
4. Then we have the Encoder as E = ⎜ 0 1 0⎟<br />
.<br />
⎜ ⎟<br />
⎝ 0 0 4⎠<br />
5. Then the encoded matrix is given by<br />
82
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
⎛ 2 5 19⎞<br />
⎛ − 2 5 76⎞<br />
⎜ ⎟ ⎛−1<br />
0 0⎞<br />
⎜<br />
⎟<br />
⎜20<br />
0 23⎟<br />
⎜ ⎟ ⎜−<br />
20 0 92⎟<br />
X = ME = ⎜ ⎟ ⎜ 0 1 0⎟<br />
= ⎜ − ⎟ .<br />
9 19 8<br />
⎜ ⎟ ⎜ ⎟ 9 19 32<br />
⎝ ⎠<br />
⎜<br />
⎟<br />
0 0 4<br />
⎝ 5 19 0 ⎠<br />
⎝ − 5 19 0 ⎠<br />
Hence the encoded numeric message is given by<br />
-2 5 76 -20 0 92 -9 19 32 -5 19 0<br />
⎛ ⎞<br />
⎜ = 1 0 0 ⎟<br />
−1<br />
−1<br />
6. Clearly the Decoder E is the given by E =<br />
⎜<br />
0 1 0<br />
⎟<br />
.<br />
⎜ 1 ⎟<br />
⎜ 0 0 ⎟<br />
⎝ 4 ⎠<br />
7. The encoded numeric message is to be decoded by first writ<strong>in</strong>g the<br />
encoded matrix X from the received message as<br />
⎛ − 2 5 76⎞<br />
⎛ ⎞ ⎛ 2 5 19⎞<br />
⎜<br />
⎟ ⎜ = 1 0 0 ⎟ ⎜ ⎟<br />
−1<br />
⎜−<br />
20 0 92⎟<br />
⎜ ⎟ ⎜20<br />
0 23⎟<br />
M = XE = ⎜<br />
⎟<br />
0 1 0 =<br />
−<br />
⎜ ⎟ ⎜ 9 19 8 ⎟ .<br />
9 19 32<br />
⎜<br />
⎟<br />
1<br />
⎜ ⎟ ⎜ ⎟<br />
0 0<br />
⎝ − 5 19 0 ⎠ ⎝ 4 ⎠ ⎝ 5 19 0 ⎠<br />
8. This matrix M is converted <strong>in</strong> to numeric message as<br />
2 5 19 20 0 23 9 19 8 5 19<br />
9. This stream <strong>of</strong> numerals is converted <strong>in</strong> to the text message as<br />
2 5 19 20 0 23 9 19 8 5 19: BEST WISHES<br />
4. 1 A word on Security:<br />
In case <strong>of</strong> us<strong>in</strong>g the standard codes one could recognize <strong>in</strong>tuitively<br />
or by any way the codes <strong>of</strong> use from the codes allotted for the alphabets. So the use <strong>of</strong><br />
codes <strong>in</strong> a random or chaotic way or by us<strong>in</strong>g some process , <strong>in</strong>creases the security level.<br />
4.1.1 Example<br />
Instead <strong>of</strong> us<strong>in</strong>g the standard codes A-1, B-2…….Z-26 and 0 for space<br />
If we use the codes assigned as<br />
A-7 , B-6 , C- 5 , D-4 , E-3 ,F-2 ,G-1 , H-8 , I-9 , J-10 , K-11 , L-12 , M-13 . N- 15 , O-<br />
16, P-17 , Q- 18 , R- 19, S-14 , T-20 , U-26 , V-25 , W-24 , X- 23 , Y -22 ,Z- 21. and 0<br />
for space.(In a random way or by us<strong>in</strong>g some generator us<strong>in</strong>g Number theory or<br />
comb<strong>in</strong>atorics)<br />
Then the message BEST WISHES is given by the matrix<br />
⎛ 6 3 14 ⎞<br />
⎜<br />
⎟<br />
⎜20<br />
0 24⎟<br />
M = ⎜<br />
⎟ Instead <strong>of</strong><br />
9 14 8<br />
⎜<br />
⎟<br />
⎝ 3 114 0 ⎠<br />
M<br />
⎛ 2<br />
⎜<br />
⎜20<br />
= ⎜ 9<br />
⎜<br />
⎝ 5<br />
5<br />
0<br />
19<br />
19<br />
19⎞<br />
⎟<br />
23⎟<br />
8 ⎟<br />
⎟<br />
0<br />
⎠<br />
83
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
Any one who <strong>in</strong>tervene the communication uses the standard codes for this<br />
message matrix will get a confus<strong>in</strong>g message like FENT XINHCN.<br />
So the messengers are advised to make use <strong>of</strong> their convenient system <strong>of</strong> codes <strong>in</strong> order<br />
to have higher security level.<br />
5. Operation on Str<strong>in</strong>gs:<br />
We def<strong>in</strong>e the operator * (The str<strong>in</strong>g addition) as usual <strong>in</strong> the case<br />
<strong>of</strong> addition <strong>of</strong> str<strong>in</strong>gs.<br />
Example: Best*wishes = Best wishes.<br />
6. Generalization<br />
Us<strong>in</strong>g this operation we decompose the messages <strong>of</strong> larger length <strong>in</strong> to<br />
messages <strong>of</strong> shorter lengths and f<strong>in</strong>ally these are co<strong>in</strong>ed to get the message <strong>of</strong> larger<br />
length.<br />
6.1 Results and Discussion<br />
Consider the Message M: MEPCO WISHES YOU ALL THE BEST.<br />
This message is decomposed <strong>in</strong> to two messages as follows,<br />
M = M1 + M2 Where M1 = MEPCO WISHES & M2 = YOU ALL THE BEST.<br />
Now for M1:<br />
1) M E P C O W I S H E S:<br />
13 5 16 3 15 0 23 9 19 8 5 19<br />
⎛13<br />
5 16⎞<br />
⎜ ⎟<br />
⎜ 3 15 0 ⎟<br />
2) T<br />
1<br />
= ⎜ ⎟ .<br />
23 9 19<br />
⎜ ⎟<br />
⎝ 8 5 19⎠<br />
3)<br />
⎛= 1 0 0⎞<br />
⎜ ⎟<br />
E = ⎜ 0 1 0⎟<br />
Such that<br />
⎜ ⎟<br />
⎝ 0 0 4⎠<br />
4) = T E =<br />
X1<br />
1<br />
T<br />
1<br />
⎛ −13<br />
⎜<br />
⎜ − 3<br />
= ⎜−<br />
23<br />
⎜<br />
⎝ − 8<br />
5<br />
15<br />
9<br />
5<br />
64⎞<br />
⎟<br />
0 ⎟<br />
76⎟<br />
⎟<br />
76<br />
⎠<br />
E<br />
−1<br />
⎛<br />
⎜ = 1<br />
=<br />
⎜<br />
0<br />
⎜<br />
⎜ 0<br />
⎝<br />
0<br />
1<br />
0<br />
⎞<br />
0 ⎟<br />
0<br />
⎟<br />
1<br />
⎟<br />
⎟<br />
4 ⎠<br />
84
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
−1<br />
5) M = X =<br />
1 1E<br />
⎛13<br />
⎜<br />
⎜ 3<br />
⎜23<br />
⎜<br />
⎝ 8<br />
5<br />
15<br />
9<br />
5<br />
16⎞<br />
⎟<br />
0 ⎟<br />
19⎟<br />
⎟<br />
19<br />
⎠<br />
6) Message 1 = 13 5 16 3 15 0 23 9 19 8 5 19<br />
M E P C O W I S H E S<br />
Now for M2:<br />
1) Y O U A L L T H E B E S T.<br />
25 15 21 0 1 12 12 0 24 8 5 0 2 5 19 20<br />
2)<br />
3)<br />
⎛25<br />
15 21⎞<br />
⎜ ⎟<br />
⎜ 0 1 12⎟<br />
⎜12<br />
0 21⎟<br />
T = ⎜ ⎟<br />
2<br />
.<br />
⎜ 8 5 0 ⎟<br />
⎜ ⎟<br />
⎜<br />
2 5 19<br />
⎟<br />
⎝20<br />
0 0 ⎠<br />
⎛= 1 0 0⎞<br />
⎜ ⎟<br />
E = ⎜ 0 1 0⎟<br />
Such that<br />
⎜ ⎟<br />
⎝ 0 0 4⎠<br />
⎛−<br />
25<br />
⎜<br />
⎜ 0<br />
⎜ −12<br />
4) X<br />
2<br />
= T2<br />
E = ⎜<br />
⎜ − 8<br />
⎜<br />
⎜<br />
− 2<br />
⎝−<br />
20<br />
15<br />
1<br />
0<br />
5<br />
5<br />
0<br />
84⎞<br />
⎟<br />
48⎟<br />
84⎟<br />
⎟ ,<br />
0 ⎟<br />
76<br />
⎟<br />
⎟<br />
0<br />
⎠<br />
E<br />
−1<br />
⎛<br />
⎜ = 1<br />
=<br />
⎜<br />
0<br />
⎜<br />
⎜ 0<br />
⎝<br />
0<br />
1<br />
0<br />
⎞<br />
0 ⎟<br />
0<br />
⎟<br />
1<br />
⎟<br />
⎟<br />
4 ⎠<br />
−1<br />
5) M = X =<br />
2 2<br />
E<br />
⎛25<br />
⎜<br />
⎜ 0<br />
⎜12<br />
⎜<br />
⎜ 8<br />
⎜<br />
⎜<br />
2<br />
⎝20<br />
15<br />
1<br />
0<br />
5<br />
5<br />
0<br />
21⎞<br />
⎟<br />
12⎟<br />
21⎟<br />
⎟<br />
0 ⎟<br />
19<br />
⎟<br />
⎟<br />
0<br />
⎠<br />
6) Message 2 = 25 15 21 0 1 12 12 0 24 8 5 0 2 5 19 20<br />
85
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
Y O U A L L T H E B E S T.<br />
Therefore the message M = M1 * M2<br />
= M E P C O W I S H E S YOU A L L T H E B E S T.<br />
6. Results Used from matrix theory<br />
The follow<strong>in</strong>g are the results used <strong>in</strong> our paper from the theory <strong>of</strong><br />
matrices available <strong>in</strong> [3]<br />
1. To any Quadratic there exists a matrix called the <strong>Matrix</strong> <strong>of</strong> quadratic<br />
form.<br />
2. The matrix <strong>of</strong> any Quadratic form is a Real Symmetric matrix.<br />
3. The Eigen values <strong>of</strong> a real symmetric matrix are always real.<br />
4. Any Quadratic form can be reduced to canonical form by means <strong>of</strong><br />
orthogonal reduction.<br />
5 The matrix <strong>of</strong> the canonical form <strong>of</strong> a Quadratic form is a diagonal<br />
matrix.<br />
6. The <strong>in</strong>verse <strong>of</strong> a diagonal matrix with a<br />
ii<br />
as entries is noth<strong>in</strong>g but the<br />
7. Conclusion<br />
scalar diagonal matrix with<br />
1 as entries.<br />
a ii<br />
1. Diagonal matrices <strong>in</strong>duced from Quadratic forms are preferred for<br />
encod<strong>in</strong>g as their <strong>in</strong>verses can be easily obta<strong>in</strong>ed.<br />
2. This provides a transaction <strong>of</strong> least amount <strong>of</strong> messag<strong>in</strong>g between the<br />
sender and the receiver. (It is sufficient to know the codes <strong>of</strong> use and the<br />
Quadratic form).Here the security is assured as only those know about the<br />
Quadratic Forms can understand the process.<br />
3. Higher order Diagonal matrices are preferred as their <strong>in</strong>verses are easily<br />
found.<br />
4. When the size <strong>of</strong> the message is too large new str<strong>in</strong>g operations may be<br />
def<strong>in</strong>ed and the message can be splitted and suitable such process<strong>in</strong>g may<br />
be carried over.<br />
5. Higher level <strong>of</strong> security can be achieved by us<strong>in</strong>g own conventional<br />
codes or codes (As <strong>in</strong> the word on Security) processed by some structure.<br />
8. Scope <strong>in</strong> further work<br />
1. Texts with all types <strong>of</strong> characters may be utilized <strong>in</strong> the study.<br />
2. Higher level <strong>of</strong> security can be enhanced by us<strong>in</strong>g structured system <strong>of</strong><br />
codes.<br />
3. Search <strong>of</strong> Diagonal matrices <strong>in</strong>duced from any other field may be done.<br />
4. Efforts can be taken for the use <strong>of</strong> any non s<strong>in</strong>gular matrix as an Encoder.<br />
86
9. Acknowledgements<br />
The authors are very grateful to Major D.Stephen Jeyapaul , Former Head , Dept<br />
<strong>of</strong> Mathematics , Thiagarajar College <strong>of</strong> Eng<strong>in</strong>eer<strong>in</strong>g , Pr<strong>of</strong>.K.P.Radhakrishnan Former<br />
Head , Dept <strong>of</strong> Mathematics , Thiagarajar College <strong>of</strong> Arts Madurai , for their constant<br />
encouragement. The author s<strong>in</strong>cerely thanks Dr.P.S.Boopathi Manickam, Dept <strong>of</strong><br />
Humanities and Mr.S.Rajaram, Dept <strong>of</strong> English for their scholarly suggestions and timely<br />
help. Above all the author dedicates this work to his ever lov<strong>in</strong>g family members<br />
particularly to his parents without whom he can’t succeed up to this level.<br />
10. References<br />
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87<br />
[1]. http://www. richland.edu / james /lecture /.../matrices/applications.html<br />
[2]. http:// aix1.uottawa.ca/~jkhoury/cryptography.htm<br />
[3]. Vasta B.S., Vasta Suchi..,Theory <strong>of</strong> Matrices.,Third edition., New Age<br />
<strong>International</strong> , India., 2010.<br />
87